\documentclass{article}

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\begin{document}

\section*{Examples}

\subsection*{Vectors}
Vectors are bolded
\begin{align*}
    & \bvec{v}
\end{align*}

\subsubsection*{Unit Vectors}
Unit vectors have a hat over them.
\begin{align*}
    & \unitvec{v}
\end{align*}
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% Note: \unitvec does not work properly with i and j unit vectors
% (see blow)
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\subsubsection*{The $\ihat$, $\jhat$, and $\khat$ Unit Vectors}
\begin{align*}
    & \ihat + \jhat + \khat
\end{align*}

\subsubsection*{Simple Cross and Dot Products}
\begin{align*}
    & \crossvec{u}{v} = \bvec{w} \\
    & \dotvec{u}{v} = c
\end{align*}

\subsubsection*{Vector Projection}
The component of $\bvec{a}$ onto $\bvec{b}$ is
\begin{align*}
    & \comp{a}{b}
\end{align*}

\subsubsection*{Vector Component}
The projection of $\bvec{a}$ onto $\bvec{b}$ is
\begin{align*}
    & \proj{a}{b}
\end{align*}

\section*{Rotation}

\subsection*{Degrees}
Degrees are not as cool as radians
\begin{align*}
    & 90\degrees = \dfrac{\pi}{2}
\end{align*}

\subsection*{2D Rotation Matrix}
To rotate a point about the origin by $\dfrac{\pi}{4}$
\begin{align*}
    & \rotateTwo{\pi/4} = \rotateTwo{45\degrees}
\end{align*}

\section{Column Vectors}
A 2D Column Vector
\begin{align*}
    & \columnVecTwo{1}{2}
\end{align*}
A 3D Column Vector
\begin{align*}
    & \columnVecThree{1}{2}{3}
\end{align*}

\section*{Matrices}

\subsection*{Augmented Matrix}

\subsubsection*{Three (centered) columns on the left,
    two (centered) on the right}

\begin{align*}
    \begin{pmatrix}[ccc|cc]
        a_{11} & a_{12} & a_{13} & c_{11} & c_{12} \\
        a_{21} & a_{22} & a_{23} & c_{21} & c_{22} \\
        a_{31} & a_{32} & a_{33} & c_{31} & c_{32}
    \end{pmatrix}
\end{align*}

\subsubsection*{Two (right-aligned) columns on the left,
    one (left-aligned) on the right}

\begin{align*}
    \begin{pmatrix}[rr|l]
        a_{11} & a_{12} & c_{1} \\
        a_{21} & a_{22} & c_{2} \\
        a_{31} & a_{32} & c_{3}
    \end{pmatrix}
\end{align*}

\subsubsection*{Works on Any Kind of Bracketed Matrix}
\begin{align*}
    \begin{bmatrix}[rr|l]
        a_{11} & a_{12} & c_{1} \\
        a_{21} & a_{22} & c_{2} \\
        a_{31} & a_{32} & c_{3}
    \end{bmatrix}
\end{align*}

\subsection*{Gaussian Elimination}
\begin{align*}
    \begin{pmatrix}[rr|l]
        0 & 1 & 1 \\
        1 & 1 & 2
    \end{pmatrix}
    \grstep{R_2 - R_1}
    \begin{pmatrix}[rr|l]
        0 & 1 & 1 \\
        1 & 0 & 1
    \end{pmatrix}
    \grstep{R_1 \swap R_2}
    \begin{pmatrix}[rr|l]
        1 & 0 & 1 \\
        0 & 1 & 1
    \end{pmatrix}
\end{align*}

\section*{Other Stuff}

\subsection*{Norm}
\begin{align*}
    & \norm{a}
\end{align*}

\subsection*{Modulus}
\begin{align*}
    & \modulus{a}
\end{align*}


\subsection*{Systems of Equations}
\begin{align*}
    & \systeme{x + y = 2, 3x + 4y = 5} \\ \\
    & \systeme{x + y + z = 5, 10x + 3y + 5z = 12, 2x + 7y + 3z = 15}
\end{align*}

\subsubsection*{This Even Works outside of math mode}
We can solve the system \systeme{x + y = 2, 3x + 4y = 5} with Gaussian
Elimination.

\subsection*{How I use Implication Arrows}
\begin{align*}
    & x + y^2 = 5 \\
    \implies & y^2 = x - 5 \\
    \implies & y = \pm \sqrt{x - 5} \\
\end{align*}

\end{document}